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Abstract

The plane of polarization of two or more wavelengths can be rotated by a predetermined amount through the use of two or more identical half-wave plates in series, the axes of which are oriented at predetermined angles with respect to the incident plane of polarization. For two wavelengths a rotation of 90° may be accomplished by the use of two plates with their slow axes at angles of 22.5°+δ and 67.5°−δ, respectively. For three wavelengths a 90° rotation is obtained by using three plates at angles of 11.25°+δ, 45°, and 78.75°−δ respectively. The quantity δ is a small angle usually less than 1° which determines the spectral range of achromatization. Identical half-wave plates are easily obtained by cutting a single splitting of mica or a plane parallel sheet of other birefringent material. The wavelength for which the plates have half-wave retardation is not critical. Rather than having a smaller angular aperture than the single half-wave plate, the three-element achromatic rotator has a larger angular aperture. The use of the stereographic projection of the Poincaré sphere for graphical solution of polarized light problems is discussed briefly.

References

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a For the two- and three-element rotators Δ1 and Δ2 were calculated by means of Eqs. (4) and (5), and were checked by constructions on the Poincaré sphere. The values agreed within ±0.5° for the two-element rotators and within ±3° for the three-element rotators. In the four-element case, Δ1, Δ2, Δ3, and Δ4 were obtained by means of the Poincaré sphere and are therefore only approximate.b The wavelengths, λ1, λ2,… were calculated assuming that the half-wave plate material had zero dispersion of birefringence and that λ0=546 mμ.cβ(Δ) is defined as the ratio of light intensity transmitted when an analyser is placed after the rotators, first parallel to and then perpendicular to the polarizer, evaluated at the wavelength for which each component plate has a retardation of Δ. The quantity β can be calculated by matrix calculus10 or from the Poincarfé sphere.11,12

a For the two- and three-element rotators Δ1 and Δ2 were calculated by means of Eqs. (4) and (5), and were checked by constructions on the Poincaré sphere. The values agreed within ±0.5° for the two-element rotators and within ±3° for the three-element rotators. In the four-element case, Δ1, Δ2, Δ3, and Δ4 were obtained by means of the Poincaré sphere and are therefore only approximate.b The wavelengths, λ1, λ2,… were calculated assuming that the half-wave plate material had zero dispersion of birefringence and that λ0=546 mμ.cβ(Δ) is defined as the ratio of light intensity transmitted when an analyser is placed after the rotators, first parallel to and then perpendicular to the polarizer, evaluated at the wavelength for which each component plate has a retardation of Δ. The quantity β can be calculated by matrix calculus10 or from the Poincarfé sphere.11,12